This project represents a broad investigation into algorithmic algebraic geometry, traversing complexity theory, Diophantine approximation, real and p-adic geometry, numerical algorithms, and the nascent field of statistical algebraic geometry. The main focus is systems of sparse polynomial equations --- a family of computational problems central in many applications. The proposed algorithms have immediate impact in certain areas of engineering where the PI and co-PIs have close connections with industry and faculty outside of mathematics. Intellectual merits include (1) an approach to a deterministic solution of Smale's 17th Problem, (2) optimal stochastic root counts for sparse polynomial systems, and (3) new algebraic examples of problems on the border between P and NP.
Broader impacts include novel approaches to problems from engineering and computer science, the training of postdoctoral researchers, and the education of graduate students. In particular, PI Rojas and graduate student Rusek have an ongoing collaboration with Sandia National Laboratories (including publically-available software) on rigorously quantifying uncertainties in the failure of physical structures, e.g., the storage of nuclear waste. Co-PI Avendano and PI Rojas also have ongoing work with Prof. Daniele Mortari (of the Texas A and M Aerospace Engineering Department) on satellite orbit design, with applications to surveillance and astronomical observation.
